Derrick's Theorem [2] states that this divergence is unavoidable for time- ...... The last identity is the non-Abelian version of the Adler-Bell-Jackiw anomaly3.
arXiv:hep-th/0010225v1 25 Oct 2000
Monopoles, Instantons and Confinement Gerard ’t Hooft, University Utrecht Saalburg, September 1999
notes written by Falk Bruckmann, University Jena (version February 1, 2008)
Contents 1 Solitons in 1+1 Dimensions 1.1 Definition of the Models . . . 1.2 Soliton Solutions . . . . . . . 1.3 Chiral Fermions . . . . . . . . 1.4 Outlook to Higher Dimensions 2 The 2.1 2.2 2.3 2.4
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Abrikosov-Nielsen-Olesen-Zumino Vortex Approach to the Vortex Solution . . . . . . . . Introduction of the Gauge Field . . . . . . . . . Bogomol’nyi Bound for the Energy . . . . . . . Gauge Topology Description . . . . . . . . . . .
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3 Magnetic Monopoles 3.1 Electric and Magnetic Charges and the Dirac Condition . 3.2 Construction of Monopole Solutions . . . . . . . . . . . . 3.3 Existence of Monopoles . . . . . . . . . . . . . . . . . . . 3.4 Bogomol’nyi Bound and BPS States . . . . . . . . . . . . 3.5 Orbital Angular Momentum for qg Bound States . . . . 3.6 Jackiw-Rebbi States at a Magnetic Monopole . . . . . .
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1 1 3 6 9
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11 11 13 15 16
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22 22 24 27 27 29 32
4 Instantons 4.1 Topological Gauge Transformations . . . . . . . . . . . . . . . 4.2 Semiclassical Approximation for Tunnelling . . . . . . . . . . . 4.3 Action for a Topological Transition, Explicit Instanton Solutions 4.4 Bogomol’nyi Bound and Selfdual Fields . . . . . . . . . . . . . 4.5 Intermezzo: Massless Fermions in a Gauge Theory . . . . . . . 4.6 Jackiw-Rebbi States at an Instanton . . . . . . . . . . . . . . 4.7 Estimate of the Flip Amplitude . . . . . . . . . . . . . . . . . I
35 35 38 39 42 44 46 48
4.8 Influence of the Instanton Angle . . . . . . . . . . . . . . . . . 50 5 Permanent Quark Confinement 5.1 The Abelian Projection . . . . . . . . . . . . . . . 5.2 Phases of the Abelian Theory . . . . . . . . . . . 5.3 A QCD-inspired Theory for the Electroweak Force 5.4 Spontaneous Chiral Symmetry Breaking in QCD .
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52 52 57 59 61
6 Effective Lagrangians for Theories with Confinement
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7 Exercises
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References
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II
Chapter 1 Solitons in 1+1 Dimensions As an introduction we consider in this chapter the easiest field theoretic examples for solitons. These are real scalar field theories in 1 + 1 dimensions with a quartic and a sine-Gordon potential, respectively. We will concentrate on physical aspects which are relevant also in higher dimensions and more complicated theories like QCD.
1.1
Definition of the Models
We investigate the theory of a single real scalar field φ(t, x) in one time and one space dimension. The usual Lagrangian (density) is, 1 1 L = φ˙ 2 − (∂x φ)2 − V (φ) (1.1) 2 2 We consider two cases for the potential. Case (a) refers to the ‘Mexican-hat’ potential, well-known from spontaneous symmetry breaking, 2 λ 2 φ − F2 (1.2) case (a): V (φ) = 4! while case (b) is the sine-Gordon model, 2πφ (1.3) case (b): V (φ) = A 1 − cos F which is an exactly solvable system. The important point about these models is that the vacuum is degenerate. In case (a) it is two-fold degenerate, φ = ±F 1
V (φ)
−F
V (φ)
φ
F
−F
(a)
0
F
φ
(b)
Figure 1.1: Potentials under consideration: the Mexican-hat (case (a), see (1.2)) and the sine-Gordon model (case (b), see (1.3)). Both have degenerate vacua and allow for non-trivial solutions.
while in case (b) we have an infinite number of vacua, φ = nF,
n∈Z
In standard perturbation theory one considers small fluctuations η around the vacua, case (a): φ = F + η case (b): φ = 0 + η,
(η ≡ φ)
and expands the potential. Here we get the usual mass term together with three and four point interactions, 1 2 2 g 3 λ 4 mη + η + η 2 3! 4! m2 ≡ λF 2 /3, g ≡ λF 1 2 2 λ 4 case (b): V (η) = m η − η + ... 2 √4! A ≡ m4 /λ √ m ≡ 2π A/F F ≡ 2πm/ λ λ ≡ 16π 4 A/F 4 case (a): V (η) =
m is the mass of the particles of the theory (we use ~ = 1) and g and λ are defined such that the three and four point vertices are proportional to √ them. Usually one has small λ and large F such that the mass λF is fixed. Notice that the mass square would be negative on the maxima 0 and n + 12 , respectively, these so-called tachyons would render the theory unstable. 2
1.2
Soliton Solutions
The degeneracy of the vacuum results in the fact that these models possess non-trivial static solutions, which interpolate (in space) between the vacua. We call them kinks or solitons. Their existence and shape are given by the Euler-Lagrange equation1 derived from L in (1.1), ∂V φ¨ = ∂x2 φ − =0 ∂φ If we think of φ(x) as x(t), this is the equation of motion of a non-relativistic , but in a potential −V . Like the energy in ordinary particle, x¨ = − ∂V ∂x mechanics, we find a first integral, d 1 ∂V 2 2 =0 (∂x φ) − V (φ) = ∂x φ ∂x φ − dx 2 ∂φ 1 (∂x φ)2 − V (φ) = const. (1.4) 2 Since we want solutions with finite energy, we have to demand that the energy density, Z ∞ 1 2 E= (∂x φ) + V (φ) dx (1.5) −∞ 2 vanishes at spatial infinity, |x| → ∞ :
∂x φ → 0, V (φ) → 0
i.e. the above constant is zero. The remaining first order differential equation can easily be solved, Z dφ (1.6) x(φ) = p 2V (φ)
In our models we can write down the solutions exactly,
1
1 case (a): φ(x) = F tanh m(x − x0 ) 2 2F case (b): φ(x) = arctan em(x−x0 ) π
For time-independent solutions we could also work in the Hamiltonian formalism. Moreover, any of the following static solutions can be transformed into a steadily moving one by a Lorentz transfomation.
3
φ(x)
φ(x) F
F
Γ ∝ 1/m x0
Γ ∝ 1/m
F/2
x
−F
x0
(a)
x
(b)
Figure 1.2: Solitonic solutions in both potentials have nearly identical shape: The transition from one vacuum value to the next one takes place around an arbitrary position x0 . It decays with a rate that is proportional to the inverse mass of the (light) particles of the theory. x0 is an arbitrary constant (of integration) due to translational invariance. Their shapes are very similiar (cf Fig. 1.2) and show the typical behaviour: (i) Solitons interpolate between two neighbouring vacua: case (a): φ(x → ±∞) = ±F case (b): φ(x → −∞) = 0, φ(x → +∞) = F (ii) The solutions are (nearly) identical to a vacuum value everywhere except a transition region around an arbitrary point x0 . The shape of the solution there is given by the shape of the potential between the vacua. Near the vacua the solution is decaying exponentially with a rate Γ ∝ 1/m. Thus φ can be approximated by a step function, for instance in case (a), φ(x) = F sgn (x − x0 )
for |x − x0 | ≫ 1/m
In terms of mechanics one could think of a particle which passes the bottom of a valley at some time t0 . It has just the energy to climb up the hill and stay there. Actually this will take infinitely long, but after a short time it has already reached a position very near the top. Of course, the particle must have been on top of the opposite hill in the infinite past. Whenever both tops have the same height such a solution exists, no matter which form V has inbetween. 4
The energy of the solution (its mass) can be computed from (1.5) and (1.4), Z ∞ Z φ∞ E = 2 ∂x φ dφ V (φ) dx = φ−∞ −∞ 2m3 /λ for case (a), = 8m3 /λ for case (b). Alternatively we can use the saturation of the Bogomol’nyi bound (cf Exercise (i)), Z 2 E= (0) + total derivative = W (φ)|φφ∞ (1.7) −∞ In both cases the mass of the soliton is given by the cube of the mass of the elementary particles divided by λ (which has dimension (mass)2 ). The soliton is very massive in the perturbative limit. That means we are dealing with a theory which describes both, light fluctuations, the elementary particles, and heavy solutions, the solitons. This mass gap supports the validity of perturbation theory (for example for tunneling). Up to now we have only considered one-soliton solutions which interpolate between neighbouring vacua. Due to the ambiguity of the square root in (1.6) there are also solutions interpolating backwards, anti-solitons. Now one could immediately imagine solutions consisting of whole sequences of solitons and φ(x)
φ(x)
F
2F x0,1
x0,2
x0,3
F
x
0
−F
x
−F (b)
(a)
Figure 1.3: Multi-Solitons are approximate solutions built of solitons and antisolitons at arbitrary positions x0,i . The sequences are strongly constrained in case (a), while in case (b) they are arbitrary.
5
anti-solitons (see Fig. 1.3). As discussed in Exercise (ii), these approximate solutions are only valid for widely separated objects |x0,1 − x0,2 | ≫ 1/m, such that we have a ’dilute gas’. Here the Mexican-hat and the sine-Gordon model differ slightly. Solitons and anti-solitons have to alternate in the first model. From a particle point of view the anti-soliton is really the anti-particle of the soliton. If the potential is symmetric, we cannot distinguish between them. The situation is like in a real scalar field theory. In the latter model we can arrive at any vacuum by choosing the right difference between the number of solitons and anti-solitons. Now solitons and anti-solitons are distinguishable and the analog is a complex scalar field theory.
1.3
Chiral Fermions
Now we investigate the (still 1+1 dimensional) system, Lψ = −ψ¯ (γ µ ∂µ + gφ(x)) ψ where ψ is a Dirac field, γµ are the (Euclidean) Dirac matrices, which we can choose to be the Pauli matrices, γ 1 = σ1 ,
γ 4 = σ3 ,
γ5 = σ2
and φ is a solution from above2 . We could say we put fermions in a soliton background or we study the consequences to the solution if we couple fermions to it. The interaction is provided by the usual Yukawa coupling. The field φ acts like a (space-dependent) mass. If φ takes the vacuum value F everywhere, then we simply have a theory with massive fermions, Lψ = −ψ¯ (γ µ ∂µ + mψ ) ψ,
mψ = gF
The energies, i.e. the eigenvalues of the (hermitean) Hamilton-Operator,3
2
H = −i∂t = iσ2 ∂x + σ3 mψ
In particular we take φ from case (a), since with the sine-Gordon model we would be forced to use a cos-interaction which is not normalisable in 4 dimensions. 3 We use the complex notation ∂4 = −i∂t .
6
E
E mψ
−mψ
(b)
(a)
Figure 1.4: The usual spectrum of massive fermions (a) is produced by a constant vacuum solution φ ≡ F . For a soliton there is one additional zero mode (b). come in pairs ±E with |E| ≥ m. The fields with the opposite energies are generated by γ 1 = σ1 , {σ1 , σ2 } = {σ1 , σ3 } = 0 ⇒ Hψ = Eψ ⇋ H(γ 1 ψ) = −E(γ 1 ψ) In the spirit of Dirac we define the vacuum to be filled with negative energy states, the particles to be excitations with E > 0 and the anti-particles to be holes in E < 0 (cf Fig. 1.4). γ 1 is the fermion number conjugation. As well-known the chiral symmetry generated by γ5 , ψ → γ5 ψ,
¯ 5 ψ¯ → −ψγ
is broken by the mass term. But as an interaction this term can be made invariant by φ → −φ which is again a solution. That means γ5 generates a state with the same energy but with φ in the opposite vacuum. Now we really want to insert a non-trivial φ and look for its spectrum. Still σ1 generates opposite energy solutions. It comes out that again they are displaced by the soliton. What about the special case of solutions of zero energy? Acting on them, H and σ1 commute, and we choose the zero modes ψ to be eigenfunctions of σ1 , ψ2 (x) 1 ψ1 (x) 1 , ψ− = √ σ1 ψ± = ±ψ± , ψ+ = √ 2 1 2 −1 7
ψ1 (x)
ln ψ2 φ x
x ln ψ1 (a)
(b) R
Figure 1.5: Zero energy solutions in a soliton background: ln ψ1,2 ∝ ± φdx (cf
(1.8)). Only one of the solutions is normalisable, the one with the full line in (a). It is localised at the soliton position which we have chosen to be 0 here (b). A similiar picture applies for the anti-soliton.
Now we have to solve (σ1 ∂x + gφ)ψ± = σ3 Eψ± = 0 which becomes ∂x ψ1,2 = ∓gφψ1,2 ,
ln ψ1,2 = ∓g
Z
φ dx
(1.8)
Knowing the general shape of the soliton φ we see immediately that the solution with the lower sign is non-normalisable, while the one which the upper sign fulfills every decent boundary condition, since it drops exponentially. Notice that ψ1 ψ2 = const. as a general property. Thus, if we add the continuum, there is an ‘odd’ (but infinite) number of solutions. For completeness we give the formula for the normalisable zero mode, −2m/mψ 1 m ψ+ = const cosh (x − x0 ) 1 2 It is strongly localized at the position x0 of the soliton (cf Fig. 1.5). For the anti-soliton −φ the solution with the lower sign is normalisable, −2m/mψ 1 m ψ− = const cosh (x − x0 ) −1 2 8
This of course agrees with the chiral transformed ψ+ , ψ+
−2m/mψ 1 m → γ5 ψ+ = const cosh (x − x0 ) σ2 1 2 −2m/mψ i m ∝ ψ− = const cosh (x − x0 ) 2 −i
These zero modes are called Jackiw-Rebbi modes [1]. We view them as soliton-fermion bound states indistinguishable from the original true soliton4 . When we quantise the theory, ψ becomes an operator with Fermi-Dirac statistics/anti-commutation relations, especially ψˆ2 = 0. ψˆ+ and ψˆ− comˆ = 0. In general this commutator inmute with the Hamiltonian: [ψˆ± , H] volves the energy, but here we have E = 0. Whether these states are filled or empty has no effect on the energy, they are ‘somewhere inbetween fermions and anti-fermions’. As we have seen they are related by γ5 . In fact Jackiw and Rebbi [1] have shown that the soliton has two states with fermion number: n = ±1/2, no spin, no Fermi-Dirac statistics! ˆ0 = n is the expectation value of the conserved charge Q these states.
1.4
R
ˆ¯ 0 ψˆ : in dx : ψγ
Outlook to Higher Dimensions
In higher dimensions the field will still like to sit in a vacuum for most of the space-time. The kinks will now be substituted by (moving) domain walls, i.e. transition regions between domains with different φ-values. Their shape will depend on the model. In any case passing domain walls will have huge physical consequences. For example in case (a) passing from φ to −φ means transforming (by chiral symmetry) matter into anti-matter. The domains themselves are related through a discrete global symmetry, namely Z2 in case (a) and Z in case (b). In other words we have two sorts of domains in case (a) and infinitely many in case (b), respectively. Concerning the chiral fermions, they are massive inside the domains and massless on the walls. The latter are localised in the direction perpendicular 4
The counting of states depends on this interpretation (cf [1]).
9
II III I
IV
Figure 1.6: The picture we expect in higher dimensions: In the domains I to IV φ sits in different vacua, which are related by a discrete symmetry. The transition takes place on domain walls. Inside the domains fermions are massive (•), on the walls they are massless (◦) and localised perpendicular to the walls.
to the domain wall. Along the domain wall we can view them as lowerdimensional Dirac fermions. This scenario has become helpful for studying fermions on the lattice.
10
Chapter 2 The Abrikosov-Nielsen-OlesenZumino Vortex 2.1
Approach to the Vortex Solution
We try to find the analogue of domain walls in 2+1 dimensions. They will come out as vortices or strings. We have to take a complex (or two component real) scalar field, φ1 ~ (2.1) φ = φ1 + iφ2 , φ = φ2 We use the generalization of (1.1) with a global U(1) invariance, L = −∂µ φ∗ ∂µ φ −
2 λ ∗ φ φ − F2 2
(2.2)
as our starting point. Notice that the vacuum manifold is now a circle |φ| = F . For the desired soliton solution we combine it with the directions in space at spatial infinity, ~ → F ~x , φ → F eiϕ |x| → ∞ : φ |x|
(2.3)
where ϕ is the polar angle in coordinate space. One such solution is depicted in Fig. 2.1. The solution could also be a deformation of this, but should go a full circle around the boundary. Since the map on the boundary is non-trivial, φ must have a zero inside. 11
Figure 2.1: A typical vortex solution with isospace vectors φ~ depicted in (2 dimensional) coordinate space ~x. The field ‘winds around once’ at spatial infinity as ~ inside, especially the position of the zero, a general feature. Angle and length of φ are still arbitrary. All these configurations are specified by the winding number 1.
But this non-trivial map at spatial infinity has the effect that the energy, Z ~ ∗ ∂φ ~ + V (φ, φ∗ ) E = d2 x ∂φ (2.4) is divergent, since the rotation of φ enters the kinetic energy, xi xj F δij − |x| → ∞ : ∂i φj → |x| |x|2 2 X F2 F2 (∂i φj )2 → (2 − 2 + 1) = |x|2 |x|2 i,j=1 Z Z ∞ F2 2 ~ ∗~ . . . log. divergent d x ∂φ ∂φ → 2π d|x| |x| 0 Thus in a theory with global U(1) invariance, there exists a vortex, but its energy (per time unit in three dimension) is logarithmically divergent! Derrick’s Theorem [2] states that this divergence is unavoidable for timeindependent solutions in d ≥ 2. Since it is only a mild divergence, the solution still plays a role in phase transitions in statistical mechanics.
12
2.2
Introduction of the Gauge Field
Now we cure the above divergence by making the U(1) invariance local in the standard manner. We add a gauge field Aµ and replace the partial derivative in (2.2) by the covariant one, ∂µ φ → Dµ φ = (∂µ − ieAµ ) φ
(2.5)
~ (we still deal with ~ the chance to converge better than ∂φ This gives Dφ ~ Since static solutions). In other words the divergence is absorbed in A. ~ will only have a component asymptotically φ depends only on the angle ϕ, A in this direction Asymptotically, φ is real at the x-axis, φ → F eiϕ |ϕ=0 = F and the gradient has only a y component, ∂r φ ∂x φ 0 ~ = 1 ∂φ → = ∂y φ ϕ=0 ∂ φ ϕ=0 iF/r r ϕ
~ from the demand of vanishing covariant derivative, We can read off A ~ ~ → 1 φ−1 ∂φ, A ie
Ax → 0,
Ay →
1 er
For the general case (at any point (x, y)) we perform a trick, namely we can rotate φ locally to be real, φ → ΩF
with Ω(~x) = eiϕ
thus, ~ −1 ~ → − 1 Ω∂Ω A ie In fact the covariant derivative vanishes asymptotically, −1 ~ ~ ~ Dφ → ∂Ω + Ω(∂Ω )Ω F = Ω∂~ Ω−1 Ω F = 0 ~ is, The general form of A 13
~ A
Φ ~ B
~ (cf (2.6)) leads to a vortex Figure 2.2: The introduction of a circular gauge field A with quantised magnetic flux Φ = n 2π e .
1 xj Ai → − ǫij 2 e r
(2.6)
As we expected it has only a ϕ-component, Ar → 0,
Aϕ →
1 er
Furthermore it is a pure gauge asymptotically and the field strength vanishes, ~ ~ → 1 ∂ϕ, A e
Fij → 0
giving a solution with finite energy per unit length. It can be shown that the choices for φ and A are solutions of the EulerLagrange equations asymptotically. If we try to extend them naively towards the origin, A runs into a singularity. Instead one could make an ansatz for φ and A and try to solve the remaining equations numerically [3]. But already from the asymptotic behaviour we can deduce a quantised magnetic flux, Z Z 2π ~ σ= ~ d~x = gm , Φ = Bd~ A gm = e S C=∂S For higher windigs we have analogously, Φ = ngm
with n ∈ Z 14
The situaton is very much like in the Ginzburg-Landau theory for the superconductor. In this theory an electromagnetic field interacts with a fundamental scalar field describing Cooper pairs. The latter are bound states of two electrons with opposite momentum and spin. As bosonic objects they can fall into the same quantum state resulting in one scalar field φ. The potential of the scalar field is of the Mexican-hat form with temperature-dependent coefficients. In the low temperature phase the symmetry is broken and the photons become massive. That means if a magnetic field enters the superconductor at all, it does so in flux tubes. Performing an AharonovBohm gedankenexperiment around such a tube leads to a flux quantum, ΦSC = ngSC The only difference to the above model is a factor 2 from the pair of electrons, qSC = 2e,
2.3
gSC =
2π π = qSC e
Bogomol’nyi Bound for the Energy
Adding the field strength term to (2.2) and (2.5), the complete Lagrangian reads, L = −Dµ φ∗ Dµ φ −
2 1 λ ∗ φ φ − F 2 − Fµν Fµν 2 4
(2.7)
The energy integral is now, Z 1 2 λ ∗ 2 ∗ 2 2 E = d x Di φ Di φ + F12 + φ φ−F 2 2 In the gauge where φ is real1 the integrand consists of a sum of squares, Z 1 2 λ ∗ 2 2 2 ~2 2 2 2 E = d x (∂i φ) + e A φ + F12 + φ φ−F 2 2 where the second and the fourth term cannot be zero at the same time. For the Bogomolnyi bound we reduce the number of squares by partial integration 1
At the origin this gauge Ω−1 = e−iϕ becomes singular, since ϕ is ambiguous at this point.
15
as for the soliton (cf (1.7) and Exercise (i)), ~ 2 φ2 = (∂i φ ± eǫij Aj φ)2 ± eφ2 F12 + total der. (∂i φ)2 + e2 A 2 √ √ 1 2 1 λ ∗ 2 2 2 2 2 ∓ λ(φ2 − F 2 )F12 φ φ−F = F12 ± λ(φ − F ) F12 + 2 2 2 Notice that the new square in the first equation looks like a covariant derivative, but it is not. The boundary contributions are easily to be calculated. For the special choice2 of √ λ = e2 mφ = mA = 2eF the energy simplifies further, Z 2 √ 1 2 2 2 2 2 2 E = d x (∂i φ ± eǫij Aj φ) + F12 ± λ(φ − F ) ± eF F12 2 Z 2 2 ≥ eF F12 d x
For the saturation of the bound the first two equations can be solved numerically, while the rest gives the total magnetic flux, E ≥ eF 2 n
πm2 2π =n 2 e e
(2.8)
Again we have found the typical dependence mass2 /coupling for heavy topological objects.
2.4
Gauge Topology Description
For the vortex as well as for the soliton we have seen that the asymptotic behaviour is important, in the sense that the requirement of finite energy forces the configurations to fall into disjoint ‘classes’. Interpolating between these classes must include configurations with divergent energy. Now we want to clarify this topological property. Along the lines of spontaneous symmetry breaking, (2.7) is a U(1) gauge theory coupled to a Higgs field φ. The vacuum manifold |φ| = F is U(1)invariant, but the special choice φ = F breaks the U(1) down to 1l: No 2
This choice corresponds to the type I/type II phase boundary of the superconductor.
16
~ A
Φ1
Φ2
Φ3
Figure 2.3: A non-trivial configuration of vortices carrying total flux (n1 + n2 +
. For topological reasons it cannot be continuously shrinked to the trivial n3 ) 2π e2 vacuum (unless n1 + n2 + n3 = 0). The total flux also results in a lower bound for 2 the energy (cf (2.8)): E ≥ |n1 + n2 + n3 | πm for λ = e2 . e2
gauge transformation leaves this special value invariant. That is, the gauge transformation leading to this gauge must be a mapping from the boundary of R2 to U(1)/1l, Ω : S 1 −→ U(1)/1l ≡ U(1) The identity has been (formally) divided out, since for the general case Ω need not come back to the same group element. It is allowed to differ by another group element belonging to the subgroup which leaves the vacuum choice invariant. We say the Higgs field φ transforms under the group U(1)/1l. The mappings from S 1 into a manifold M themselves form a group, called the first homotopy group π1 (M). π1 measures the non-contractibiliy of M, i.e. the ‘existence of holes’. For contractible M all mappings are identified and π1 is simply the identity. The Lie group U(1) itself is a circle S 1 . The first homotopy group of S 1 is well-known to be the group of integers, π1 (S 1 ) = Z Notice that this group is Abelian. Thus from topological arguments each vortex carries a quantum number Q ∈ π1 (U(1)/1l) = Z 17
B
D
X A
C
Figure 2.4: In a general Higgs theory G → G1 each vortex represents an element of the group G2 = π1 (G/G1 ). The fusion rules are governed by this group, which might be non-Abelian. For the depicted case we have gA gB = gX = gC gD . which can be identified with the total flux number n. We have found an abstract reasoning for the quantisation of this physical quantity. There are infinitely many U(1)/1l vortices and they are additively stable. Other situations may occur. Let a group G be spontaneously broken down to a subgroup G1 , Higgs G −→ G1 Then in the same spirit G2 = π1 (G/G1 ) is the group of vortex quantum numbers. Whenever G2 is non-trivial G2 6= 1l, there are stable vortices. Their fusion rules are given by the composition law of the group G2 , which in general might be non-Abelian. Then the quantum number of the vortex is not additive, and one vortex cannot ‘go through the other one’ without leaving a third vortex (cf Fig. 2.4, 2.5 and Exercise (iii)). This situation plays a role in the theory of crystal defects, where the vortices go under the name of ‘Alice strings’. Generically it does not occur in the Standard Model of elementary particle physics. But let us consider a ‘double Higgs theory’, SU(2)
Higgs Higgs −→ U(1) −→ Z2 I=1 I=1 18
A
B
C B
A
Figure 2.5: Alice strings: When a non-Abelian group G2 is associated to the vortices, the hitting of two of them (A and B) will lead to a third one C = ABA−1 B −1 as indicated by the dashed contour. The SU(2) theory is broken by a Higgs field in the adjoint (I = 1) representation down to the maximal Abelian subgroup U(1). This U(1) ∼ = SO(2) corresponds to the residual rotations around the preferred vacuum direction. Afterwards the theory is broken down further to Z2 by another adjoint Higgs field. Z2 as the center of SU(2) is mapped onto the identity in the adjoint representation (cf (2.10)) and thus acts trivially on the Higgs field. To be explicit we parametrise SU(2) as a three-sphere (cf Fig. 2.6), SU(2) ∋ Ω = a0 1l + iai σi ,
aµ real,
a20 + ~a2 = 1
(2.9)
The center Z2 sits on the poles a0 = ±1, ~a = 0, Ω = ±1l. Clearly it sends an I = 1 field φ back to itself, φ →Ωφ = Ω† φ Ω = (±)2 1lφ1l = φ
(2.10)
It is just the identification of opposite points that leads to the group SO(3), SU(2)/Z2 ∼ = SO(3) In addition non-contractible closed paths are created, namely those which connect two opposite points. The first homotopy group of SO(3) is nontrivial, π1 (SO(3)) = Z2 Therefore, a SU(2)/Z2 vortex carries a multiplicative quantum number ±1. +1 stands for the contractible situation, which is homotopic to the trivial 19
a0 Ω = 1l
a2 , a3
a1
Ω = −1l
Figure 2.6: The group SU (2) parametrised as a three-sphere a20 +
P3
2 i=1 ai
= 1. The center Z2 = ±1l sits on the poles, and every closed path can be contracted to a point: π1 (SU (2)) = 1l. After identification of opposite points (×) one arrives at the group SO(3). Every path connecting two opposite points is now closed, but not contractible: π1 (SO(3)) = Z2 .
vacuum. Unlike the case above there is only a finite number of different vortices, namely two. Another significant difference to the U(1)/1l case is the orientability: One could try to label the quantum numbers of the vortices by arrows. But as Fig. 2.7 indicates, these arrows are unstable in the SU(2)/Z2 case. Altogether we have that the U(1) → 1l vortex has an additive quantum number n ∈ Z and is orientable. while the SU(2) → U(1) → Z2 vortex has a multiplicative quantum number ±1 ∈ Z2 and is non-orientable. This statement has a very interesting physical consequence (cf Fig. 2.8). Imagine two U(1) → 1l vortices with flux 2π/e, respectively. The total flux is 4π/e. But seen as SU(2) → U(1) → Z2 vortices the intermediate vortex is equivalent to the vacuum with flux zero. The vortices have snapped creating 20
−
−
+
+
Figure 2.7: A graphical proof that the SU (2)/Z2 vortex is non-orientable: Two incoming vortices with quantum number −1 produce a vortex with quantum number +1. It is equivalent to the vacuum, and the arrows become inconsistent.
a pair of something that carries magnetic charge. We conclude there must be magnetic monopoles with magnetic charge 4π/e (or an integer multiple of it). We will analyse these magnetic monopoles in the next chapter.
U (1) → Z2
SU (2) → U (1) → Z2
Figure 2.8: The snapping of vortices (see text).
21
Chapter 3 Magnetic Monopoles 3.1
Electric and Magnetic Charges and the Dirac Condition
Studying the vortices of Chapter 2 automatically revealed the existence of pure magnetic charges in non-Abelian gauge theories G → U(1). As worked out, I=1
SU(2) → U(1) produces magnetic monopoles with magnetic flux ±4π/e = gm . The minimally allowed electric charge is q = e/2 for I = 1/2 doublets. Indeed the Dirac condition, qgm = 2πn
n∈Z
is exactly obeyed. We remind the reader of its origin. In Maxwell’s theory isolated magnetic sources are excluded and the magnetic field is the curl of a smooth gauge field. Thus for a monopole the Maxwell field has to be singular at the so-called Dirac string. This is a curve which extends from the monopole to infinity1 and carries a magnetic flux gm . Physically the magnetic monopole is the endpoint of a tight magnetic solenoid which is too thin to detect. The string itself can be moved to a different position by a singular gauge transformation. 1
It could also extend to a second monopole with inverse charge such that the net flux through a surface including both vanishes.
22
D′
D I
II
C
M
Figure 3.1: For introducing a magnetic monopole (M) into an Abelian theory, a Dirac string (D) is needed. When moving around the string on a circle C the wave function picks up a phase. This phase is proportional to the magnetic flux carried by the string, and the Dirac condition follows. The Dirac string can be put on a different position (D′ ) by a singular gauge transformation. In the fibre bundle construction the circle C is the overlap region of two patches (I,II).
Now we consider a matter field in the presence of that string. The vector potential enters the Schr¨odinger equation via the conjugate momentum, ~ V) H = H(~p − q A, When we go around the string the wave function picks up a phase, Z ~ d~r = qgm q A C
In the Aharonov-Bohm effect the same consideration leads to a phase shift of two electron beams. Since the wave function has to be single-valued and no AB effect shall take place, we have the restriction that qgm = 2πn. The existence of one monopole quantises all electric charges.2 One can avoid the singularities by a fibre bundle construction: Every two-sphere around the monopole consists of two patches on which the gauge fields are regular, respectively. The patches overlap on some circle C around the string. There a gauge transformation (‘transition function’) Ω = eieΛ connects the fields,
2
~ (II) = A ~ (I) + ∇Λ, ~ A
ψ (II) = ψ (I) eieΛ
Introducing the magnetic charge qmag = gm /4π the Dirac condition reads qqmag =
n/2.
23
For Ω to be single-valued Λ has to fulfil, q[Λ(ϕ = 2π) − Λ(ϕ = 0)] = 2πn The functions Λ fall into disjoint classes, the simplest representatives of which are just proportional to the angle ϕ around the string, Λ=
n ϕ q
On the other hand the magnetic flux is given by the A-integral on the boun~ dary. Here it is the ∇Λ-integral on that circle, Z ~ r = Λ|2π = 2πn/q gm = ∇Λd~ (3.1) 0 C
The Dirac condition has a topological meaning: The transition function Ω : S 1 → U(1) has a winding number and (3.1) is how to compute it.
3.2
Construction of Monopole Solutions
After the excursions through lower dimensions we present in this section a 3+1 dimensional theory. No surprise, it is a non-Abelian gauge theory with gauge group SU(2) and a Higgs field φ in the I = 1 representation, 2 1 a a 1 λ 2 L = − (Dµ φa )2 − φa − F 2 − Fµν Fµν 2 8 4
(3.2)
Both φ and A are elements of the Lie algebra su(2) ∼ = R3 , φ = φa τa ,
Aµ = Aaµ τa ,
τa = σa /2
and the non-Abelian definition of the covariant derivative and the field strength includes commutator terms, Dµ φa = ∂µ φa + ǫabc Abµ φc ,
a = ∂µ Aν − ∂ν Aµ + ǫabc Abµ Acν Fµν
We look for static solutions of the field equations. Repeating the arguments from the previous chapter we expect φ to live on a sphere with radius F asymptotically: φa φa = F 2 . Topologically it is a mapping from S 2 (as the boundary of the coordinate space) to another S 2 (of algebra elements with 24
Figure 3.2: The Higgs field of a monopole configuration shows a ‘hedgehog’ behaviour. It points in the same direction (in isospace) like its argument (in coordinate space) and has winding number 1.
fixed length). The degree of this mapping is an integer. Alternatively one can see immediately that φ transforms under (SU(2)I=1 ≡ SO(3))/U(1). Its second homotopy group3 is the group of (even) integers. The one to one mapping, √ φa (x) → F xˆa , xˆi = xi /|~x|, |~x| = xi xi , i = 1, 2, 3 is the first non-trivial mapping. On the boundary the same things happen as before, φ ‘winds around once’. Notice that the isospace structure (indices a) is mixed with the space-time structure (indices i). Inside, φ is of the same form, φa (x) = φ(|~x|)ˆ xa
(3.3)
with a regular function φ(|~x|). The solution is depicted in Fig. 3.2. It is called a ‘hedgehog’ and has a zero inside. Corresponding to our previous discussions we make the natural ansatz: A0 = 0,
Aai = ǫiaj xˆj A(|~x|)
(3.4)
The first condition means that there is no electric field, the second one is the analogue of the circular gauge field in (2.6). Again it exploits the mixing of isospace and coordinate space indices. The magnetic field at spatial infinity looks like if there were a magnetic charge inside: Bi ∝ xi /|~x|3 . 3
Analogously to π1 , the second homotopy group π2 is the group of mappings from S 2 into the given manifold.
25
What happens after spontaneous symmetry breaking? To extract the physical content one usually makes use of the local gauge symmetry. We diagonalise φ, i.e. force it to have only a third component. The corresponding gauge is called ‘unitary gauge’, 0 0 0 0 φ→ + (3.5) F η F and η are the vacuum expectation value and the fluctuations of the Higgs field, respectively. The third component of the Higgs field gets a mass, √ Mη ≡ MH = F λ For the gauge fields it is the other way round. A1µ and A2µ are massive vector bosons, A3µ is the massless photon referring to the unbroken U(1) in the third direction, MA1,2 ≡ MW ± = eF,
MA3 = 0
The gauge field coupling is denoted by ‘e’, since this is also the charge unit with respect to the residual Maxwell potential A3µ . It can be shown that under spontaneous symmetry breaking the hedgehog configuration turns into a Dirac monopole. It resides in the origin, while the Dirac string is placed along the negative z-axis. The last point is not difficult to explain (see also Exercise (iv)): The gauge transformation has to rotate φ onto the positive z-axis in the algebra. It can be written in terms of the spherical coordinates θ and ϕ. The latter becomes ambiguous on the z-axis. This does not matter at its positive part, since φ is already of the desired form there. But on the negative part it points just in the opposite direction and there are a lot of rotation matrices. Whatever direction we choose for the spontaneous symmetry breaking in (3.5), a singularity occurs: the unitary gauge changes the asymptotic behaviour of φ from the hedgehog to the trivial one. Like in the explicit case the singularity is always situated on the opposite part of the chosen axis. We have found again that the existence of the Dirac string is gauge invariant, its position is gauge dependent.
26
3.3
Existence of Monopoles
What is the general feature of theories which allow for monopole solutions? The U(1)em must be embedded as a subgroup4 in a larger non-Abelian group G, and π1 (G) < Z For the winding number to be finite, G must have a compact covering group5 . This is the topological reason for the statement, that there are no magnetic monopoles in the electroweak sector of the Standard Model SU(2)I × U(1)Y → U(1)em U(1) has a non-compact covering group, namely R+ . Thus π1 (SU(2)I × U(1)Y ) is still Z and vortices refuse to snap. In Grand Unified Theories (GUT) the Standard Model is embedded in a larger group like SU(5). Then monopole solutions become possible again and have magnetic flux 2π/e. As we will see in the next section, its mass is bigger than the mass of the massive vector bosons W of the theory. The GUT scale is 1016 GeV and the monopole mass is of the order of mPlanck . So far no experiment has detected magnetic monopoles. Perhaps they exist somewhere in the universe. Not only GUT’s but also cosmological models predict their existence. The generalization to monopoles with added electric charge was introduced by Julia and Zee [4]. These particles are called ‘dyons’. It is easy to imagine that one can add multiples of A± to a monopole, gm =
3.4
4π , e
q = ne
Bogomol’nyi Bound and BPS States
For estimating the monopole mass we again use the Bogomol’nyi trick, Z 1 ~ λ 2 1 ~2 3 2 2 2 E= dx (Dφa ) + (φa − F ) + Ba 2 8 2 4
It is also possible to embed a product of U (1)’s into G which will be the case for the Abelian Projection of SU (3) and higher groups in chapter 5. 5 The covering group of a given Lie group is constructed from the same Lie algebra, but is simply connected.
27
Note that the homogeneous field equation for non-Abelian theories read, 1 a Di Bia = ǫijk Di Fjk =0 2 It is the usual Bianchi identity that allows for the introduction of the A-field. We use it to reduce the number of squares, ~ a )2 + B ~ a2 = (Dφ ~ a±B ~ a )2 ∓ 2 B ~ a Dφ ~ a (Dφ We rewrite the last term in a total derivative, ~ B ~ a Dφ ~ a = ∂( ~ a φa ) B Its contribution to the energy is gauge invariant, and we compute it in the unitary gauge, Z Z 3 ~ ~ ~ 3~n = 4π F d x∂(Ba φa ) = F B e 2 S∞ Thus the energy of the monopole is bounded from below by the mass of the W -boson, Z 1 ~ 4π λ 2 3 2 2 2 ~ E = d x (Dφa ± Ba ) + (φa − F ) + 2 MW (3.6) 2 8 e The bound is saturated for vanishing potential, λ = 0. The exact solution to the remaining equations, ~ a±B ~ a = 0, Dφ
|φ| → F
was given by Sommerfield and Prasad. These so-called BPS states have a mass, Mmon =
4π MW e2
and are important for supersymmetric theories.
28
3.5
Orbital Angular Momentum for qg Bound States
In this section we look for electrically charged particles bound to the monopole. All particles with U(1) charge originate from SU(2) representations, I = integer −→ qU (1) = ne 1 1 I = integer + −→ qU (1) = (n + )e 2 2 consistent with the Dirac condition, gmon =
4π −→ qgmon = 2πn e
Let us take a minimal charge q = 12 e, i.e. a field ψ in the defining representation of SU(2), 1 ψ1 I= : ψ= 2 ψ2 and consider ψ near a monopole. The wave equation reads: D 2 ψ + µ2 ψ → 0
or
(γν Dν + µ) ψ → 0
µ plays the role of a binding potential. In the regular description the monopole solution has φa (x) = φ(|x|)ˆ xa . Its rotational symmetry can only be exploited a if we rotate φ together with ~x. That is spacial SO(3) rotations must be coupled to isospin SU(2) rotations, SU(2)space × SU(2)isospin −→ SU(2)diag where SU(2)diag is the invariance group of the monopole. The representation of our ψ in this SU(2)diag is, Ltot = Lspace + Lisospin 1 ltot = lspace ± 2 ψ may be a scalar under spacial rotations but carries now half spin! Similar things happen, when we give ψ an ordinary spin ± 21 : the angular momentum 29
D
el
mon (a)
(b)
(c)
Figure 3.3: An electrically charged particle (el) feels the Dirac string (D) of a magnetic monopole (mon). When moving around the string, ψel picks up a phase factor (a). This is equivalent to move the monopole around the electric charge with its electric string (b). For bound states (c) we choose both strings to point in opposite directions.
becomes an integer! We have found that [5] q = 21 e particles will bind to a magnetic monopole = 2π in such a way that with gm = 4π e q the orbital angular momentum is integer + 12 . The spin becomes half-odd integer, although the monopole is a spin 0 object. The anomalous spin addition theorem for qg bound states with q · g = 2π reads: integer + integer −→ integer+ 21 etc. Something like this was never seen in quantum field theory before. The reasoning heavily relies on the existence of Dirac strings. Imagine an electric charge in the fundamental representation and a monopole like in Fig. 3.3. The wave function of the electric charge ψel feels the string coming from the monopole. The Maxwell equations allow us to interprete the resulting phase shift also after interchanging electric and magnetic charges. Then ψmon feels the string coming from the electric charge. Accordingly, the eg bound state has two strings. When they are oppositely oriented, the bound state looks like if it has a string running from −∞ to ∞. We remind the reader that one part of the string is only felt by the magnetic monopole wave function, while the other part only by the electric charge wave function. 30
1
2
2
1
=
(?)
=
= 1 (a)
2
2
1
1
2
(b)
2 (c)
1
Figure 3.4: The two-particle-wave function of monopoles (a) and electric charges (b) is symmetric due to the fact that the charges do not feel string of their own kind. What will happen for bound states (c) is discussed in the text.
Now consider two identical bosonic monopoles and two identical bosonic electric charges. Since the charges do not feel strings of its own kind, they can be moved around freely (cf Fig. 3.4(a) and 3.4(b)). In the same way combine the charges into identical bound states. What is their statistics? What happens to the wave function when we interchange two of these? They are the states of Fig. 3.4(c), and, considering the wave functions of these objects, with all strings attached, there will be no anomalous sign switch if we interchange the two objects. However, we may now observe that, as long as the objects remain tightly bound, each as a whole feels a string that runs from −∞ to +∞: since they carry both electric and magnetic charge, they each feel the combination of the strings from Fig. 3.4(a) and 3.4(b). To be precise: if ~r1 is the center of mass of bound state 1 and ~r2 is the center of mass of bound state 2, the wave function is, ψ12 (~r1 , ~r2 ) = ψcm (
~r1 + ~r2 )ψrel (~r1 − ~r2 ) 2
and it is ψrel (~r1 − ~r2 ) that feels a Dirac string running through the origin from z = −∞ to z = +∞. The point is now that we may remove this Dirac string by multiplying ψrel with eiϕ(~r1 −~r2 ) This produces a minus sign under the interchange ~r1 ↔ ~r2 . The bound states obey Fermi-Dirac statistics [6]! After the Dirac string is removed, the system 31
of two identical bound states is treated as an ordinary system of particles such as molecules.
3.6
Jackiw-Rebbi States at a Magnetic Monopole
In the first chapter we have seen that there are chiral fermions in the background of a kink. We briefly discuss this effect for the monopole. We introduce fermions ψ transforming under some representation of the gauge group SU(2) given by the generators (T a )ij , ¯ L = Lmon − ψγDψ − Gψ¯i φa Tija ψj The first part Lmon is the theory (3.2) we have discussed so far. Since the fermion couples to the Higgs field it gets a mass, 0 unitary gauge: hφa i → 0 : mψ = C · GF F where C is a coefficient depending on the representation. The energies are again the eigenvalues of the Hamilton operator, γ4
∂ → γ4 E i∂t
We use an off-diagonal representation for the matrices α ~ and β, 0 1l 0 ~σ , γ4 = β, β = −i γ4~γ = −i~ α, α ~= −1l 0 ~σ 0 The energy equation reads h i ~ a ) + βGT a φa ψ = Eψ α ~ (~p + gT a A We split ψ into its chirality components ψ = monopole in its regular form ((3.3),(3.4)),
χ+ χ−
and insert the magnetic
[~σ (~p + gA(|~x|)T a (~σ ∧ ~r)a ) ± iGφ(|~x|) T a xˆa ] χ± = Eχ∓ 32
which for E = 0 separates into equations for χ+ and χ− , respectively. We have already discussed the symmetry of this equation. Invariant rotations are generated by the total angular momentum J~, ~ +S ~ + T~ J~ = L ~ S ~ and T~ are the ordinary angular momentum, the spin and the where L, isospin, respectively. Let us work in the defining representation t = 1/2 ~ = 0, (q = ±e/2) and look for the simplest solutions with J~ = L ~ + T~ = 0 S For this case Jackiw and Rebbi found one solution, E = 0,
(j = l = 0).
Note that j = 0 inspite of s = 1/2. As for the kink, the Jackiw-Rebbi state lies inbetween the fermion and anti-fermion eigenstates. Whether it is full or empty does not change the energy of the system (Fig. 3.5). In most cases the baryon number is just a conserved charge, the monopole (anti-monopole) contributes to since it has: baryon number electric charge
= ± 1/2 = ± e/4
E empty
full or empty
full Figure 3.5: The spectrum of fermions in the background of a magnetic monopole. There are E = 0 Jackiw-Rebbi states, the degeneracy of which depends on the total angular momentum j.
33
Much more is to be said about the electric charges. Here we only state that the bound states behave like particles or anti-particles under U(1)charge . For the adjoint representation t = 1 we have j = 1/2. There are now two solutions with E = 0 and jz = ± 1/2 and we get a 22 -fold degeneracy. Notice that the Jackiw-Rebbi solution is a chiral wave function: χ+ and χ− are eigenstates of γ5 , which is block-diagonal in the chosen representation.
34
Chapter 4 Instantons New topological objects, the so-called instantons, arise in pure non-Abelian gauge (Yang Mills) theories in four dimensions. We approach the topic by investigating the structure of gauge transformations.
4.1
Topological Gauge Transformations
Let us work in the Weyl (=temporal) gauge A0 = 0, where the theory reduces to 1 1 ~ 2 ~ 2 1 Ea − Ba . F0ia = ∂0 Aai , L = (∂0 Aai )2 − Fija Fija = 2 4 2 The Lagrangian density is nothing but the difference of kinetic and potential energy in a Yang Mills sense. The action of a gauge transformation Ω on a gauge field A is, 1 Aµ → Ω(x)( ∂µ + Aµ )Ω−1 (~x) ie Obviously the surviving invariance of the gauge A0 = 0 consists of timeindependent gauge transformations, ∂t Ω = 0 ⇒ Ω(~x, t) = Ω(~x) just like a global symmetry in time. The Hamiltonian of the theory, Z Z 1 3 a a ~2 + B ~ 2) d3~x(E H = d ~x(Ei ∂0 Ai − L) = a a 2 35
S3
R3 ∪ {∞}
Figure 4.1: The stereographic projection identifies the three-sphere with the threespace compactified at spacial infinity, which is the image of the north pole.
is the sum of the kinetic and potential energy, and commutes with these gauge transformations, [H, Ω(~x)] = 0 We can diagonalise both operators simultaneously, H|ψi = E|ψi,
Ω(~x)|ψi = ω(~x)|ψi
The eigenvalues ω are constants of motion. Now infinitesimal gauge transformations give rise to eigenvalues λ, Ω(~x) = 1l + iǫΛ(~x),
Λ(~x)|ψi = λ(~x)|ψi
The only values of λ consistent with the unbroken spacial Lorentz transformations is λ(~x) = 0 However, a class of Ω(~x) exists that cannot be obtained from infinitesimal gauge rotations Λ(~x). We remind the reader of the stereographic projection, which identifies the three-space R3 compactified at spacial infinity with the three-sphere S 3 (Fig. 4.1). If Ω has the same limit when going to spacial infinity in any direction, it can be regarded as a function on R3 ∪ {∞} ∼ = S3. Since SU(2) is again a three-sphere we have, Ω : S3 → S3 36
E[Ai (λ, ~x)]
Ω1 λ=0: A
λ
λ=1:
Ω1A
0
1
(a)
λ
(b)
Figure 4.2: A continuous line of gauge fields connects two gauge equivalent configurations (a). Since the intermediate points are physically different, they have different energies (b) and tunnelling is expected.
Similiarly to vortices and monopoles, these mappings are classified by the third homotopy group, which for SU(2) is an integer, π3 (SU(2)) = π3 (S 3 ) = Z Again the one to one mapping Ω1 is distinct from the trivial mapping Ω0 (~x) ≡ 1l and has winding number one. Representatives of higher windings are delivered by raising this function to the nth power, Ωn (~x) = (Ω1 (~x))n Still these operators can be diagonalised together with the Hamiltonian. Since they are unitary, their contants of motion are characterised by an angle θ, Ω1 (~x)|ψi = eiθ |ψi,
Ωn (~x)|ψi = einθ |ψi,
θ ∈ [0, 2π)
(4.1)
θ is a Lorentz invariant. It is called the instanton angle. It is a fundamental parameter of the theory, which could be measured in principle1 . Although Ωn (~x) form topologically distinct gauge transformations, they act on the space {Ai (~x)} which is topologically trivial. Consider a continuous line of gauge fields connecting two gauge equivalent A’s (Fig. 4.2(a)),
1
Ai (~x) → Ai (λ, ~x),
Ai (1, ~x) =
Ω1
Ai (0, ~x)
The experimental evidence that there is little CP violation in QCD indicates that θ must be very small or zero.
37
But at fractional λ this is not a gauge transformation. These gauge fields lie on different orbits, i.e. are physically different! So do their energies, i.e. the expectation value of H in these configurations, E[Ai (λ, ~x)] = h{Ai(λ, ~x)}|H|{Ai(λ, ~x)}i If λ = 0 and λ = 1 are vacua, the energy is higher inbetween as drawn in Fig. 4.2(b). The system may tunnel through the gauge transformation Ω1 . How one actually computes the tunnelling rate and how the action enters this calculation will be explained in the next section.
4.2
Semiclassical Approximation for Tunnelling
For the eigenfunctions ψ of the Hamiltonian H of an ordinary one dimensional quantum mechanical system Hψ = Eψ,
1 H = p2 + V (x) 2
(~ = m = 1)
we write formally, pψ = −i
p ∂ ψ = 2(E − V (x))ψ ∂x
Thus Z p 2(E − V (x)) dx) ψ ∝ exp(i is an approximate solution, i.e. describes the leading effects (in ~). In the classically allowed regions E > V (x) the wave function just oscillates, while in the forbidden regions there is an exponential suppression, Z p E < V (x) : ψ ∝ exp(− 2(V (x) − E) dx) We deduce that the following quantity approximates the tunnelling amplitude, Z Bp exp(− 2(V (x) − E) dx) A
38
V (x)
E x A
B
Figure 4.3: The semiclassical situation for tunnelling through a potential barriere (see text).
where A and B are the boundary points of the forbidden region V (A) = V (B) = E. The sign switch V − E → E − V is equivalent to p → ip, p2 → −p2 or to, t → it = τ,
E → iE,
V → iV
The first replacement means that we can interchange the meaning of ‘allowed’ and ‘forbidden’ by going to an imaginary time. For field theories one passes from Minkowski to Euclidean space, accordingly. Moreover, the integral can be rewritten as the action for imaginary times, Z Bp Z tB Z τB 2(V (x) − E) dx = p x˙ dt = L(τ ) dτ = Stot (if E = 0) A
tA
τA
Thus the dominant contribution to a tunnelling transition is obtained by computing the action of a classical motion in Eulidean space, and write, e−|Stot |
(4.2)
For tunnelling in the space of gauge fields we are automatically driven to the following topic.
4.3
Action for a Topological Transition, Explicit Instanton Solutions
Let us seek for a tunnelling configuration along the lines of Fig. 4.4. In the infinite (Euclidean) past the gauge field is trivial A = 0. Then it evolves 39
~ = Ω1 (~x) 1 ∂~ Ω−1 (~x) A 1 ig
x4 → +∞
1111 0000 0000 1111 0000 1111 0000 1111
~ ~x) A(λ,
A=0
A=0
A=0
x4 → −∞
Figure 4.4: A tunnelling process in Yang Mills theory. A trivial vacuum at x4 → −∞ evolves into a vacuum with winding number 1 at x4 → +∞.
~ = Ω1 (~x) 1 ∂~ Ω−1 (~x) in somehow and arrives at the first non-trivial vacuum A 1 ig the infinite future. During the whole process A should vanish at the spacial boundary. For x4 → +∞ we already know this, since Ω1 (~x) becomes constant there. But now we can write A as a pure gauge on the whole boundary of R4 , 1 ~ −1 Ω1 (~x) at x4 → +∞, ~ A → Ω1 (x) ∂ Ω1 (x) with Ω1 (x) = const. elsewhere. ig A gauge equivalent (now we leave A4 = 0), but more symmetric way is to choose Aµ → Ω1 (x)
1 ∂µ Ω−1 1 (x) ig
(4.3)
with Ω1 (x) →
x4 1l + ixi τi , |x|
|x| =
√
xµ xµ
Notice that Ω1 lives on the boundary of R4 which is a three-sphere. It has the same degree as discussed above and mixes coordinate space and isospace. The last point will be crucial for finding explicit instanton solutions. The problem becomes simpler due to the higher symmetry. The action of Ω1 on a fundamental spinor is, 1 x4 + ix3 1 = Ω1 (x) (4.4) −x2 + ix1 |x| 0 40
and covers the whole sphere. The symmetry is such that an SO(4) rotation in Euclidean space is linked to isospin SU(2) rotations, SO(4) ∼ = SU(2)L ⊗ SU(2)R SU(2) → SU(2) ⊗ 1l Obviously the Lie algebra so(4) is 6=3+3 dimensional. For a matrix α ∈ so(4), αµν ∈ R,
αµν = −ανµ
we define the ‘dual transform’ α ˜ as, 1 α ˜ µν := ǫµνρσ αρσ 2 The six degrees of freedom can be divided as follows, αµν = 6 =
1 (α 2
+ α) ˜ µν + 3 +
1 (α 2
− α) ˜ µν 3
˜˜ ≡ α) and The terms on the right handside are selfdual and anti-selfdual (α correspond to representations of su(2)L and su(2)R , respectively. A field ψ a transforming as an I = 1 representation under SU(2)L can be written as, a ψ a = ηµν aµν
The coefficients are denoted by the tensor η. It is selfdual, a a ηµν = η˜µν
and of course anti-symmetric in (µ, ν), as easily seen from the explicit representation, ηija = ǫaij ,
a ηi4 = δia ,
a η4i = −δia
As ǫiaj in three dimensions it provides the mixing of coordinate space and isospace. The η-tensor can now be used to describe the vector field Aaµ in the adjoint representation. One finds from (4.3) and (4.4) that, asymptotically, Aµ → Ω1
1 a xν ∂µ Ω−1 τa 1 ≡ 2ηµν ig |x|2 41
It becomes singular when approaching the origin. Which smoothened connection near the origin minimises the action? With our knowledge we try, a Aaµ = ηµν xν A(|x|)
Indeed, the profile, A(|x|) =
2 |x|2 + ρ2
(4.5)
makes the action minimal, 1 S= 4
Z
a a Fµν Fµν =−
8π 2 g2
(4.6)
This number has to be exponentiated in (4.2). If |g| is small, the resulting rate is very very small. Furthermore since its expansion for small g gives zero in all orders, tunnelling processes will not be seen in pertubation theory. The new length ρ is the width of the profile. Since these configurations are local events in space and time, they are called instantons or pseudoparticles. The action is independent of ρ, i.e. we have found a whole manifold of instanton solutions. This so-called moduli space also contains the position zµ of the center of the instanton which was chosen to be at the origin in above.
4.4
Bogomol’nyi Bound and Selfdual Fields
The instanton fulfills a Bogomol’nyi bound. We write Z Z 1 1 a a 2 a ˜a ˜ (Fµν − Fµν ) + Fµν Fµν −S = 8 4
(4.7)
The number of squares has reduced from 3 · 6 in (4.6) to 3 · 3. To see that the second term is a total derivative needs some effort, 1 a ˜a 1 a a Fµν Fµν = ǫµνρσ Fµν Fρσ 4 8 g g 1 ǫµνρσ (∂µ Aaν + ǫabc Abµ Acν )(∂ρ Aaσ + ǫade Adρ Aeσ ) = 2 2 2 1 g2 = ǫµνρσ (∂µ Aaν ∂ρ Aaσ + gǫade ∂µ Aaν Adρ Aeσ + ǫabc ǫade Abµ Acν Adρ Aeσ ) 2 4 42
The g 2 term vanishes because of the symmetry of the δ’s in (b, c, d, e) together with the anti-symmetry of ǫ in (µ, ν, ρ, σ). Similar symmetry arguments give the following result2 , 1 a ˜a 8π 2 Fµν Fµν = 2 ∂µ Kµ 4 g
(4.8)
with the Chern-Simons current g abc a b e g2 a a ǫ (A ∂ A + ǫ Aν Aρ Aσ ) (4.9) Kµ = µνρσ ρ ν σ 16π 2 3 being a gauge variant quantity. The asymptotic behaviour of this current, |x| → ∞ :
Kµ →
1 xµ 2π 2 |x|4
gives the following surface integral Z 8π 2 8π 2 1 1 8π 2 −S = 2 d3 σK⊥ = 2 |x|3 area(S13 ) 2 3 = 2 g g 2π |x| g 3 S∞
(4.10)
The vanishing of the square in (4.7) means that the field strength is selfdual. From (4.5) we compute a Fµν
4 a ρ2 a ˜ = Fµν = − ηµν g (|x|2 + ρ2 )2
and indeed, Dµ Fµν (≡ Dµ F˜µν ) = 0. In general, the Bogomol’nyi bound is a useful tool to solve the Yang-Mills equations. After having introduced the A-field, one needs to solve Dµ Fµν = 0. This equation corresponds to the inhomogeneous Maxwell equation and therefore is second order in A. The demand for selfdual fields Fµν = F˜µν is only first order in A. Now the Yang-Mills equation is automatically fulfilled because of the Bianchi identity Dµ F˜µν = 0. 2 For all configurations the second term in (4.7) is a multiple of − 8π . It is g2 a topological quantity, called the Pontryagin index. Since the integral can be reduced to the surface, it corresponds to the winding number Ω1 : S 3 → S 3 discussed above. 2
For readers familiar with differential forms we give the following equivalent equation: trF ∧ F ∝ d tr(A ∧ dA − 2ig 3 A ∧ A ∧ A) with a proper definition of the wedge product for algebra elements
43
4.5
Intermezzo: Massless Fermions in a Gauge Theory
The coupling of fermions to the gauge field is done in a standard way by the vector current, ¯ µψ Jµ = ψγ where we dropped the isospace structure. For massless fermions, the axial current is conserved, too, ¯ µ γ5 ψ, Jµ5 = ψγ
∂µ Jµ = ∂µ Jµ5 = 0
Denoting by JµL and JµR the projections onto γ5 eigenstates, we can write, Jµ = JµL + JµR ,
Jµ5 = JµL − JµR
Thus the total number of fermions as well as the difference of left-handed and right-handed fermions are classically conserved. In order to look whether these statements survive the quantisation of the theory, consider the matrix element h0|Jµ5 |ggi g are the gauge photons (gluons) which couple to Jµ , not to Jµ5 . The corresponding lowest order Feynman diagram is a one-loop graph depicted in Fig. (4.5). We do not want to go into the details of the calculation, but rather sketch the Dirac matrix structure, Γµαβ (k, p, q) ∝ Tr γµ γ5 k1 Jµ5 ∝ γµ γ5
(γ, k1 ) (γ, k2 ) (γ, k3 ) γα γβ k12 k22 k32
γα
p
(4.11)
g
k2 k3 γβ
q
g
Figure 4.5: The lowest order Feynman graph leading to the chiral anomaly. 44
ki and (p, q) are the momenta of the fermions and the gauge photons, respectively (k + p + q = 0). The diagram is totally symmetric in the sense that in (4.11) we can put γ5 also after γα or γβ due to the anti-commutation relations. But the diagram is linearly divergent, and the infinity must be regularised. We prefer the introduction of Pauli-Villars mass terms, but the result will be independent of the regularisation method, ΓPV µαβ (k, p, q) ∝ Tr γµ γ5
M − i(γ, k1 ) M − i(γ, k2 ) M − i(γ, k3 ) γα γβ k12 + M 2 k22 + M 2 k32 + M 2
The symmetry is lost by renormalisation, namely the finite part of the diagram will depend on where one puts γ5 . The ambiguity in γ5 is removed by the following choice, pα Γµαβ (k, p, q) = qβ Γµαβ (k, p, q) = 0 kµ Γµαβ (k, p, q) ∝ ǫαβγδ pγ qδ 6= 0 The gauge invariance due to the two gauge photons has survived, but Jµ5 is not conserved anymore, ∂µ h0|Jµ (x)|ggi = 0 ∂µ h0|Jµ5 (x)|ggi =
g2 a ˜a h0|Fµν Fµν |ggi 16π 2
3 The last identity is the non-Abelian version of the Adler-Bell-Jackiw . R 4 a a anomaly 2 32π ˜ The topological density enters here, remember that d xFµν Fµν = g2 for R an instanton. It effects the charges Q5 = d3 xJ05 (x) in the way that the charge ‘after the instanton’ (at x4 → −∞) differs by two from the charge ‘before the instanton’ (at x4 → +∞), Z d4 x ∂µ Jµ5 = Qafter − Qbefore =2 5 5
One fermion has flipped its helicity from right to left. In other words, the instanton adds a left-handed particle and removes a right-handed anti-particle (the other way round for right-handed particles). 3
The Adler-Bardeen theorem guaranteees that there are no effects in higher order pertubation theory.
45
instanton
Left
Right
Figure 4.6: The (interaction with the) instanton flips the helicity of the fermion from right to left as shown in the text.
4.6
Jackiw-Rebbi States at an Instanton
How to understand the fact, that the interaction with an instanton flips the helicity of the fermion (Fig. 4.6)? Let us investigate the gauge group SU(2) with fundamental fermions (I = 1/2). As we know the spinorial group SU(2)L ⊗ SU(2)R couples to the gauge group SU(2)L , SU(2)L ⊗ (SU(2)L ⊗ SU(2)R ) For left-handed and right-handed fermions we have, 2L × (2L × 1R ) = 3L + 1L ,
2L × (1L × 2R ) = 2L × 2R
respectively. There is one state with jL = jR = 0 which indeed has a normalisable solution in four-space, ψ=
const. (|x|2 + ρ2 )3/2
This Jackiw-Rebbi state is a chiral eigenstate and fulfils the (Euclidean) Dirac equation γDψ = 0 ~ ~x) connecting two vacua in the gauge Let us come back to the line A(λ, A4 = 0 (Fig. 4.2(a)) and choose just x4 as the parameter of the configuration, λ ≡ x4 46
E E>0 instanton anti-inst. E
